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Math Help - Trigonometric Integrals

  1. #1
    Junior Member
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    Trigonometric Integrals

    I'm having trouble finding out how to solve this integral.



    I know the correct answer is: 0.235619, but I want to know if my work is correct. So far, whenever I work it out, I don't get the right answer.

    From the integral,

    ((1/2)(1-cos10x))^2

    = (1/4) integral (1-cos10x)^2

    =(1/4) integral (1-2cos10x * (cos10x)^2

    =(1/4) integral 1-2cos10x * ((1/2)1+cos20x)

    From there, I plugged in the numbers, but I'm not getting the correct answer. Help?
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Kimmy2 View Post
    I'm having trouble finding out how to solve this integral.



    I know the correct answer is: 0.235619, but I want to know if my work is correct. So far, whenever I work it out, I don't get the right answer.

    From the integral,

    ((1/2)(1-cos10x))^2

    = (1/4) integral (1-cos10x)^2

    =(1/4) integral (1-2cos10x * (cos10x)^2

    =(1/4) integral 1-2cos10x * ((1/2)1+cos20x)

    From there, I plugged in the numbers, but I'm not getting the correct answer. Help?
     \sin ^4 5x = \frac{(1-\cos 10x)^2}{4} = \frac{1-2\cos 10x +\cos ^2 10x}{4} = \frac{1-2\cos 10x + \frac{\cos 20x +1}{2} }{4}

    \sin ^4 5x = \frac{2 - 4\cos 10x + \cos 20x +1 }{8}

    \sin ^4 5x = \frac{3 -4\cos 10x + \cos 20x }{8}

    when you integrate it you will get

    \frac{3\pi}{5\cdot 8}= \frac{3\pi}{40}
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  3. #3
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    Could you please explain how you arrived to your integral?

    Thanks!
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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Kimmy2 View Post
    Could you please explain how you arrived to your integral?

    Thanks!
    \int _{0}^{\frac{\pi}{5}} \sin ^4 5x \cdot dx

    \int_{0}^{\frac{\pi}{5}} \frac{3-4\cos 10x + \cos 20x }{8} \cdot dx

    \left(\frac{1}{8}\right) \left( 3x - \frac{4\sin 10x}{10} + \frac{\sin 20x}{20} \right)\mid^{\frac{\pi}{5}}_{0}

    \left(\frac{1}{8}\right) \left( \frac{3\pi}{5} - \frac{4\sin 2\pi}{10} + \frac{\sin 4\pi}{20} \right)-\left(\frac{1}{8}\right) \left( 3(0) - \frac{4\sin (0)}{10} + \frac{\sin (0)}{20} \right)

    \left(\frac{1}{8}\right) \left( \frac{3\pi}{5}\right)
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